Orientation relationships, orientational variants and the embedding approach

A statistical approach is developed for handling some standard problems involving orientation relationships. The approach is based on directional statistics.


Introduction
Phase transformations within polycrystalline materials often induce a transformation of the texture, i.e. of the statistical and spatial distribution of the crystallographic orientations of the crystallites. This usually involves the replacement of each parent crystal by several child crystals, possibly in a different symmetry group. Several mathematical models for describing these transformations have been constructed on theoretical grounds. Such models include the orientation relationships of Kurdjumow-Sachs (Kurdjumow & Sachs, 1930), Nishiyama-Wassermann (Wassermann, 1933, 1935Nishiyama,1934) and Pitsch (1959Pitsch ( , 1967. The presence of two symmetry groups leads to the problem of multiplicity of variants, i.e. orientations of (unseen) parent crystals giving rise to orientations of given child crystals. (A precise mathematical definition is given in Section 2.3.) Much work has been carried out developing methods for fitting these models and examining their adequacy, notably by Humbert and co-authors (e.g. Humbert et al., 1994Humbert et al., , 1995Humbert et al., , 2015 and by Cayron and his collaborators (e.g. Cayron et al., 2006;Cayron, 2019). Even within the topic of the austenite-martensite transformation there are very many papers in the materials science literature on problems involving variants, e.g. Nolze (2004b), Kitahara et al. (2006), Miyamoto et al. (2009), Abbasi et al. (2012Abbasi et al. ( , 2014, Koumatos & Muehlemann (2017) and Nyyssö nen et al. (2016Nyyssö nen et al. ( , 2018. These problems have been addressed by Mainprice et al. (1990) and Morales et al. (2018), for instance, in a geological context.
As far as we are aware, previous work on fitting orientation relationships and identifying variants takes the traditional viewpoint of estimating the relationship and assessing its fit, either informally or by making use of an arbitrary threshold of some measure of goodness of fit. For example, Nolze (2004aNolze ( ,b, 2005 measures the discrepancy between estimated and theoretical versions of a parameter [A] 1, 2 of an orientation relationship [see equation (1) below] by the (minimal) misorientation angle between representative rotations. The discrepancy is considered significant if it is larger than the accuracy of the experimental apparatus. In this paper we establish a sound statistical viewpoint on various types of problem involving orientation relationships and variants. In the traditional viewpoint, the key idea is that of being 'numerically close' and the typical question is 'how far apart are the measured and theoretical quantities?'. From the statistical viewpoint, the key idea is that of being 'statistically close' (in a precise probabilistic sense) and the typical question is 'what is the probability (under some hypothesis) that the measured and theoretical quantities are (at least) this far apart?'. If this probability is small then the hypothesis is rejected. We implement the statistical viewpoint here by using the embedding approach to crystallographic orientations introduced by Arnold et al. (2018), which is an application of the embedding approach to directional statistics [see e.g. Sections 2.1 and 9.1 of Mardia & Jupp (2000)]. The necessary measures of distance are constructed using orientation representations which incorporate the relevant symmetries, reducing the need to search for optimal representations within equivalence classes. The use of explicit coordinate systems is avoided.
The key points of our approach are as follows: (i) Regarding each object such as a crystallographic orientation as a single object (equivalence class) in a quotient space, rather than considering a single representative orientation.
(iii) Avoiding arbitrarily chosen explicit coordinate systems (such as are used in asymmetric domains).
The notation used in this article is more mathematical and general than that traditionally used for discussing variants. The reasons for this choice are (i) the concepts of orientation relationship and variant are appropriate in contexts wider than the crystallographic, and (ii) for the embedding approach our notation is particularly convenient. We use the word 'random' in the statistical sense of 'not deterministic', so that repeating measurement of e.g. rotations will not produce exactly the same values (rather than implying that these rotations are uniformly distributed).
Some of the material in this paper has been presented in a more mathematical form suitable for statisticians by Arnold & Jupp (2019) and Arnold et al. (2018Arnold et al. ( , 2021. Section 2 gives a mathematical description of orientation relationships and variants. The embedding approach to crystallographic orientations is recalled in Section 3 and extended to the context of variants. Sections 4, 5, 6 and 7 treat methods for estimation (sometimes referred to as 'calculation', 'determination' or 'evaluation') of orientation relationships, assessment of the adequacy of a single orientation relationship, assessment of the common parentage of child crystals and reconstruction of parents, respectively. In each of these sections established methods are recalled and then methods based on the embedding approach are given. Some practical illustrations of these new methods are given in Section 8.

Orientations of symmetrical objects
The orientation of a rigid object in R 3 can be described by a rotation that transforms it into some standard orientation. If the object is asymmetrical then this rotation is unique, so that the orientations of the object correspond to elements of the group SO(3) of rotations of R 3 . If the object has symmetry group K then a rotation U has the same effect as the rotation UH for any H in K. Then the orientations of the object correspond to elements of the space SO(3)/K, i.e. the set of equivalence classes of elements of SO(3) under the right action of K. For U in SO(3) we shall denote the corresponding left coset of K, i.e. the equivalence class {UH :

Orientation relationships
In many contexts, e.g. transformations involving a phase change, interest lies in the relationship between two random orientations with possibly different symmetry groups. Let [U] 1 and [V] 2 be random orientations in SO(3)/K 1 and SO(3)/K 2 , respectively, where K 1 and K 2 are the symmetry groups. Particularly simple models relating [U] 1 and [V] 2 are the orientation relationships. These have the form where R and A are in SO (3 In crystallography it is usual to identify K 1 \SOð3Þ=K 2 with an asymmetric domain, i.e. a connected subset of SO(3) that (apart from a set of measure zero) contains exactly one rotation in each equivalence class. Construction of asymmetric domains is considered in Section 6.3 of Morawiec (2004). Because, in any given context, there is no standard asymmetric domain, we prefer not to use such domains. An orientation relationship (1) gives rise to four types of problem: research papers (ii) The single orientation relationship problem of assessing whether or not a single orientation relationship can describe the data adequately.
(iii) The sibling problem of determining whether or not elements [V 1 ] 2 , . . . , [V n ] 2 of SO(3)/K 2 arise from some unknown common parent [U] 1 under a known orientation relationship (R, [A] 1, 2 ). If [V 1 ] 2 , . . . , [V n ] 2 are distinct then they arise from some common U if and only if they are part of a set of variants (see Section 2.3).
(iv) The reconstruction problem of estimating the unknown [U] 1 on the basis of a known (R, [A] 1, 2 ) and observations These problems are considered in Sections 4, 5, 6 and 7, respectively.

Variants
In general, an orientation relationship (1) does not yield a unique map from SO(3)/K 1 to SO(3)/K 2 . Instead, to each [U] 1 in SO(3)/K 1 it assigns s distinct elements of SO(3)/K 2 , where H 1 , . . . , H s 2 K 1 and K A being the intersection group defined as  (2) of variants can be identified with the coset space K 1 /K A .
As pointed out by Nolze (2008), in contrast to the commonly used theoretical orientation relationships [A] 1, 2 , measured orientation relationships ½ b A A 1;2 are 'irrational' in that their descriptions in terms of crystallographic planes and directions do not have low Miller indices. This is because it is not possible to make measurements with perfect accuracy, and so the orientations [U i ] 1 and [V i ] 2 are random. The argument in Proposition 1 in Appendix A shows that the number of variants associated with a measured orientation relationship is |K 1 |. In real materials the variants are usually present in unequal quantities, a phenomenon known as variant selection, and some variants may even be absent. In many contexts it is of interest to estimate the overall proportions of the variants that are present. Whereas the orientational variants considered in this paper are cosets of K A in K 1 , the types of variant introduced by Cayron (2016, 2019) in connection with the physical mechanisms underlying martensitic crystallography are cosets of other subgroups of K 1 that describe the distortion and stretch of crystal lattices.
As pointed out by Cayron (2006), the important algebraic structure associated with the set (2) of variants is that of a groupoid, i.e. a set of arrows endowed with a partially defined associative composition in which every arrow has an inverse. Denote the set (2) of variants by V. Each triple (½RUH i A 2 , . It is useful to combine the arrows into equivalence classes called operators. Each operator can be written as K A H À1 i H j K A for some i, j (but, in general, i and j are not unique) and can be regarded as being a theoretical transformation that takes the variant [RUH i A] 2 to [RUH j A] 2 . Thus, the set of operators can be identified with the double coset space The composition table of the groupoid yields a multi-valued composition on the set of operators. This provides a way of attacking the sibling and reconstruction problems described in points (iii) and (iv) at the end of Section 2.2. Gey & Humbert (2003) pointed out that, in the case of the Burgers orientation relationship (K 1 = O, K 2 = D 6 ), the number of misorientations between the variants is less than the number of variants.
is uniformly distributed on SO(3)/K. For the crystallographic groups C 1 , C 2 , C 3 , C 4 , D 2 , D 6 , T and O some useful functions t are given explicitly in Table 1. Together t and hÁ, Ái lead us to ktð½UÞ À tð½VÞk 2 ¼ htð½UÞ À tð½VÞ; tð½UÞ À tð½VÞi ð5Þ as a new measure of squared distance between elements [U] and [V] of SO(3)/K. By design it incorporates the symmetry group K and it replaces the need for misorientation angles. A useful summary of [U 1 ], . . . , [U n ] is their sample mean, ½U, which is defined as the element of SO(3)/K that minimizes P n i¼1 ktð½UÞ À tð½U i Þk 2 or, equivalently, maximizes htð½UÞ; P n i¼1 tð½U i Þi. For K = C r or D r explicit approximations to sample means are given in Section 2.3 of Arnold & Jupp (2019).
Embeddings are discussed further in Appendix B.

Embedding K 1 \SO(3)/K 2
The double coset spaces K 1 \SOð3Þ=K 2 are even more complicated than the coset spaces SO(3)/K, but the embedding approach can be used for them also.
Any t : SO(3)/K 2 ! E, where E is a vector space, can be averaged over K 1 to give a corresponding t : K 1 \SOð3Þ=K 2 ! E, defined by We shall exploit such t in order to carry out inference on orientation relationships. If t has properties (ii) and (iii) of Section 3.1 then so does t. On the other hand, t having property (i) (being one-to-one) does not imply that t has this property also. In a few very special cases (e.g. K 1 and K 2 isomorphic to C 2 and C 3 and with a common axis) t is identically zero. In the context of phase transitions it seems that such symmetries are not of practical interest.

Estimation based on orientations of parents and children: established methods
Suppose that we are given paired observations ([U 1 ] 1 , of random orientations that satisfy the orientation relationship (1). The problem is that of estimating the unknown [A] 1, 2 (and R if it is not known). In the case K 1 = K 2 = C 1 (i.e. no symmetry, so that variants do not occur), an explicit estimate of A was given by Mackenzie (1957). In the general case, it follows from (1) Here 'local' is used in the sense that the estimate uses only quantities with index i, which in many cases means that they are measured at the ith location.
Using this approach Nolze found that, for transformation from face-centred cubic to body-centred cubic lattices, the standard theoretical models did not provide a good fit to the experimental data that he was investigating

Estimation based on orientations of parents and children: embedding approach
The embedding approach uses a suitable embedding Thus, [A] 1, 2 can be taken as the sample mean of the images by t 2 [defined by (6) ng. An alternative method of estimation uses the R and [A] 1, 2 that maximize S 1 ðR; ½A 1;2 ; fð½U i 1 ; ½V i 2 Þg n i¼1 Þ, which is defined by  Table 1 Some embeddings t K : The symmetric arrays r u i and N r are defined in equations (37) and (38), respectively.
Then a 100 Â (1 À )% bootstrap confidence region for [A] 1, 2 is For the corresponding test of H 0 : [A] 1, 2 = [A 0 ] 1, 2 (where [A 0 ] 1, 2 is a specified theoretical [A] 1, 2 ), the p-value (i.e. the probability under H 0 of another sample producing a value of ½ b A A 1;2 at least as extreme as that observed) is The hypothesis H 0 is rejected at significance level if p or, equivalently, if the estimated ½ b A A 1;2 lies outside the 100 Â (1 À )% bootstrap confidence region (11) for [A] 1, 2 .
A flow chart for implementation of the techniques described in this subsection is given in Fig. 1.

Estimation based on orientations of children alone: established methods
In some settings, no observations on the orientations of parent grains are available. Nevertheless, in many cases it is possible to estimate  [V 3 ] 2 ). These pairs are taken to obey (at least approximately) a form of the orientation relationship (1), so that for some U 1 , U 2 , U 3 and A 1 , A 2 , A 3 . It is assumed that U 1 ' U 2 ' U 3 ' U and A 1 ' A 2 ' A 3 ' A for some U and A in SO(3) (assumptions that are reasonable if the orientations are measured at points near a triple junction and the local orientation relationships vary only slowly with position). It then follows from (13) that for i, j in {1, 2, 3} it is possible to Then or, equivalently, where H m ¼ H À1 j H i . For given i and j, solutions for A of the equation obtained from (16) by replacing approximate equality by exact equality are far from unique. On the other hand, for general V 1 , V 2 , V 3 (outside some set of measure 0), the three corresponding equations as i, j run through distinct (unordered) pairs in {1, 2, 3} do have a unique solution. In view of this, Humbert et al. (2015) suggested estimating the rotation A by any element of SO(3) that minimizes X ði;jÞ Since H m , e H H i and e H H j are not known, Nyyssö nen and coworkers recommend averaging the above b A A as H m , e H H i , e H H j run through K 1 , K 2 , K 2 . Note that averaging of rotations is not well defined.

Estimation based on orientations of children alone: embedding approach
Given an embedding t 2 : the sum being over ordered pairs (i, j) of distinct elements of {1, . . . , n}. A flow chart for implementation of the technique described in this subsection is given in Fig. 2.

Assessing the adequacy of a single orientation relationship
Whereas some data sets can be fitted well by the orientation relationship (1), for others such a single orientation relationship is inadequate and several orientation relationships are required. For data sets in which a single orientation relationship does not suffice, it is possible to identify clusters of observations within each of which the data share a single orientation relationship.

Established methods
We are unaware of anything in the crystallographic literature that considers exactly this problem.

Embedding approach
One way of exploring the adequacy of (1) is to use cluster analysis to divide the n pairs of observations ([U 1 ] 1 , [V 1 ] 2 ), . . . , ([U n ] 1 , [V n ] 2 ) into subsets, each of which consists of pairs that give similar estimates of [A] 1, 2 . Cluster analysis is described in detail by Everitt et al. (2011), who give several algorithms. We find it convenient to use the following simple divisive algorithm. For an estimate ½ b A A 1;2 found using the method of Section 4.2 and for i = 1, . . . , n, define d i by where is defined in (9). The maximum possible value of d i is [which does not depend on U in SO (3) Placing all observations with d i close to 2 into a cluster, reestimate ½ b A A 1;2 for that cluster. Repeat this process on the remaining observations, thus grouping them sequentially into clusters having similar local orientation relationships. If only one cluster is found then the single orientation relationship ½ b A A 1;2 will describe the data well. Whereas the clustering of locations used by Johnstone et al. (2020) and Ostapovich & Trusov (2021) in the construction of crystal orientation maps takes place on SO(3)/K and is based on misorientation angles, the clustering used here takes place on K 1 \SOð3Þ=K 2 and is based on the d i of (22). The clustering of locations used by Gomes de Araujo et al. (2021) in the reconstruction of parent microstructure is based on misorientation angles between adjacent grains.
A flow chart for implementation of the technique described in this subsection is given in Fig. 3.

Common parentage
In some contexts, such as locating the boundaries of parent grains, it is useful to be able to assess whether or not a set of child crystals are from the same parent. Observed elements [V 1 ] 2 , . . . , [V n ] 2 of SO(3)/K 2 can be considered as arising from some common parent if and only if they are close to variants corresponding to some element of SO(3)/K 1 .

Established methods
The method of Gey & Humbert (2003) and Karthikeyan et al. (2006) Table 2 Values of the squared radius 2 . 2 is defined in equation (23).
r odd 1 + 2 1Àr r 2 r even 1 þ r 2 2 1Àr f1 þ 2 À1 r r=2 g À r 2 =ðr þ 1Þ D 2 2 D r (r ! 3) r odd 2 1Àr r 2 r even r 2 2 1Àr f1 þ 2 À1 r r=2 g À r 2 =ðr þ 1Þ T 32/9 O 6/5 retical misorientation angles. These observed misorientation angles are where the maximum is over e V V i in SO (3) with ½ e V V i 1;2 = ½V i 1;2 for i = 1, . . . , n. Cayron et al. (2006) respectively, (at least approximately) and with O ik as some value of the composition of O jk with O ij . Such a 'nucleus' triple is 'grown' by adding progressively further grains, to obtain a set of grains in which each triple is coherent. This is continued until a maximal such set is obtained.

Embedding approach
The embedding approach uses a suitable embedding t 2 : SO(3)/K 2 ! E. Divisive cluster analysis is then applied to t 2 ([V 1 ] 2 ), . . . , t 2 ([V n ] 2 ) using as a measure of squared distance [see (5) A flow chart for implementation of the technique described in this subsection is given in Fig. 4.

Reconstruction
In some contexts the orientations [V 1 ] 2 , . . . , [V n ] 2 of child crystals are observed but the parent crystals are not visible. The reconstruction problem is that of estimating the orientation [U] 1 of the parent crystal, assuming a given orientational relationship of the form (1) (14) holds. It follows that If R and A are known then applying this method to the three pairs (i, j) obtained from three variants yields a unique value for [U] 1 . If observations on n variants are available then Humbert & Gey (2002) and Gey & Humbert (2003) recommend that the values of [U] 1 given by the n 3 À Á triples of variants be averaged. Note that averaging of rotations is not well defined.

Embedding approach
One reconstruction method using the embedding approach starts with a suitable embedding t 2 of SO (3)    A flow chart for assessing common parentage.
where A is any representative in SO(3) of [A] 1, 2 . In (28) it can be assumed without loss of generality that H 1 = I 3 , the identity. Locating the maximum of (28) involves maximization over SOð3Þ Â K n 1 , so it may be useful to consider an alternative estimator which is easier to compute. The left-hand approximate equality in (14) gives and so for any embedding t 1 of SO(3)/K 1 . Define A À1 by Then (30) gives Thus it is reasonable to estimate [U] 1 by the sample mean (based on t 1 ) of ½R À1 V 1 A À1 1 ; . . . ; ½R À1 V n A À1 1 , i.e. the [U] 1 that maximizes A flow chart for implementation of the technique described in this subsection is given in Fig. 5.      The data set considered by Wendler et al. (2017) consists of 9707 pairs of orientations ([U i ] 1 , [V i ] 2 ), i = 1, . . . , 9707, of austenite and martensite, respectively. For each pair the orientations are measured at sites that are close but separated by a boundary between grains. The locations of the pairs on the surface of a steel sample are shown in Fig. 6. Fig. 7 displays these orientations on two stereonets (stereographic projections), one for the (face-centred cubic) austenite phase and the other for the (body-centred cubic) martensite phase. The symmetry groups K 1 and K 2 are both equal to the octahedral group O. In Fig. 7, each disc represents the upper half of the unit sphere, and each orientation [U i ] 1 or [V i ] 2 is represented by the three points at which the three (unordered) orthogonal axes determined by [U i ] 1 or [V i ] 2 intersect the upper half of the unit sphere. In each diagram there is considerable variation, which is due partly to the differing crystal orientations [U i ] 1 of the austenite phase prior to the transformation. However, there is a degree of clustering present, which can be explained by differing orientation relationships [A] 1, 2 among the pairs of observations. A cluster analysis, as outlined in Section 5, using S 1 from (8) [rather than S 2 from (7)] was carried out. It reveals six clusters in O\SOð3Þ=O of values ½ b A A 1 1;2 ; . . . ; ½ b A A 6 1;2 of ½ b A A 1;2 . Most observations belong to cluster 1 (n 1 = 7214) or cluster 2 (n 2 = 1844). Note that this is not a spatial clustering but rather a clustering of common orientation relationships. The locations of the clusters are shown by the colouring of the symbols in Fig. 6, and all six clusters occur at locations spread all over the surface of the steel sample. Fig. 8(a) shows the estimated orientations ½ b A A c 1;2 for each of the six clusters, c = 1, . . . , 6. These same orientations are shown together with the ½U T i V i 1;2 in Fig. 8(b). In drawing diagrams of this type we have had to make the choice to display each ½ b    SO(3)/O, (arbitrarily) selecting the representation closest to the first observation ½U T c1 V c1 2 in each cluster. Fig. 7(b) appears to be a left-right reflection of the centre plot [labelled '{100}') in Fig. 5(d) of Wendler et al. (2017)]. (That the figures are not identical may be due to a difference in convention regarding Euler angles.) The colour codings in the two figures are completely different; that in Fig. 7(b) is derived from having taken full account of all ambiguities in rotation.

Testing whether [A] 1, 2 has a given value
In the analysis in Section 8.1 the estimated orientation relationship ½ b A A 1;2 for cluster 2 is very close to the identity I 3 . We can test the hypothesis H 0 : [A] 1, 2 = [I 3 ] 1, 2 by forming a bootstrap confidence region for ½ b A A 1;2 using the methods of Section 4. For demonstration purposes, we took a random subset of 50 observations from cluster 2. We then generated B = 100 bootstrap samples from this subset, and reestimated [A] 1, 2 in each case. The distribution of values of Fig. 9(a), and the similarity measure 0 for the hypothesized value falls well outside the distribution, a convincing rejection of H 0 . Fig. 9(b) displays the original estimate ½ b A A 1;2 and the 100 replicate estimates, all lying well away from the hypothesized value [I 3 ] 1, 2 .

Software
The analyses in Section 8 were performed using the freely available statistical software R (https://www.R-project.org/). A general and flexible MATLAB tool kit for the analysis of crystallographic data is provided by the open-source crystallographic toolbox MTEX (Bachmann et al., 2010).

Conclusions
The intrinsic symmetries of crystal structures provide a challenge to statistical analysis due to the ambiguities of their representations. The challenge becomes greater in a setting where the orientations of crystals with different symmetries are to be compared.
In this paper we have applied the embedding approach from directional statistics to the representation of crystallographic orientation data. This approach enables us to reformulate standard problems of estimation and inference in the crystallographic setting, eliminating the ambiguities which arise from the crystallographic symmetries, while retaining the genuine multiplicity of crystallographic variants.
Although the embedded objects we work with may be unfamiliar, they are straightforward to implement in software, and should provide a practical tool for researchers seeking to characterize crystallographic structures and their transformations.

APPENDIX A The number of variants
the number of variants is s = |K 1 |/|K A |.
Proposition 1. There is a subset B of SO(3) that has measure 0 and such that if A is outside B then s = |K 1 |.
Proof. For given H i in K 1 andH H j in K 2 , define the set B ij by Then either (i) B ij ¼ SOð3Þ or (

APPENDIX B Some embeddings of SO(3)/K
The embedding approach is based on embeddings, i.e. well defined equivariant one-to-one functions t K : SO(3)/K ! E, where E is an inner-product space on which SO(3) acts, such that t K ([U]) has expectation 0 if [U] is uniformly distributed on SO(3)/K. Here, we focus on a simple choice of embedding t K of SO(3)/K into an appropriate space of symmetric arrays, where K is one of the crystallographic groups C 1 , C 2 , C 3 , C 4 , D 2 , D 6 and O or the tetrahedral group T (in Schoenflies notation). Appropriate embeddings t K for general point groups K of the first kind are considered by Arnold & Jupp (2019) and Arnold et al. (2018Arnold et al. ( , 2021. It is convenient to describe the orientation of a crystal by a frame, meaning a set of vectors or axes that are fixed in the crystal. The presence of symmetry under group K means that such a frame is equivalent to one obtained from it by a rotation in K. The equivalence classes are known as K-frames. For K = C 1 the frame is taken to be (u 1 , u 2 , u 3 ) with u 1 , u 2 , u 3 orthogonal unit vectors; for K = C 2 the frame is (u 0 , AE u 1 ) with u 0 a unit vector and AE u 1 an axis orthogonal to u 0 ; for K = D 2 it is a pair of orthogonal axes (AE u 1 , AE u 2 ); for K = C r with r ! 3 or K = D 6 with r ! 3, the vectors (u 1 , . . . , u r ) are unit normals research papers to the sides of a regular r-gon or hexagon; for O and T, the axes (AE u 1 , AE u 2 , AE u 3 ) and (AE u 1 , AE u 2 , AE u 3 , AE u 4 ) are unit normals to the sides of the cubic crystal and tetrahedral crystal, respectively. The K-frames will be denoted by square brackets, e.g. for K = C r [u 1 , . . . , u r ] denotes the K-frame arising from (u 1 , . . . , u r ). The frames used are listed in Table 3. Note that the frames are specified entirely by vectors and axes; they do not depend on any coordinate system. For notational simplicity, we shall sometimes denote a K-frame by [U].
The embeddings t K are given explicitly in Table 1. They are based on symmetric r-way arrays, r u i , which can be thought of as rth powers of vectors. In mathematical terms, they are coordinate representations of r-fold tensor products. For vectors u 1 , . . . , u r with u i = (u i, 1 , u i, 2 , u i, 3 ) T for i = 1, . . . , r, the r-way array r u i has (j 1 , . . . , j r )th entry ð r u i Þ j 1 ;...;j r ¼ Y r k¼1 u i;j k ; 1 j 1 ; . . . ; j r 3; ð37Þ e.g. 2 u i ¼ u i u T i . Some of the t K also involve the arrays N r for r even. These symmetric r tensors are defined by where kvk denotes the length of v. In coordinate terms they have entries ðN r Þ j 1 ;...;j r ¼ 1 where AE r is the group of permutations of {1, . . . , r} and ij = 1 for i = j, ij = 0 otherwise. Thus, e.g. ðN 4 Þ j 1 ;j 2 ;j 3 ;j 4 = ( j 1 j 2 j 3 j 4 + j 1 j 3 j 2 j 4 + j 1 j 4 j 2 j 3 ) /3. Some of the embeddings involve subtraction of constant terms to ensure that E{t K ([U])} = 0 under uniformity. In the case of C r , the embeddings contain oriented vectors u 0 to specify the orientation of the directed axis of symmetry.
For our purposes it is not necessary to construct the representations t K ([U]) explicitly; it is sufficient to compute inner products ht K ([U]), t K ([V])i as required, where hÁ, Ái is the standard inner product on the relevant space of symmetric arrays. The inner products are listed in Table 4.
Some much more general classes of embeddings than the t K given in Table 1 are considered by Hielscher & Lippert (2021) and Arnold et al. (2021).
In general, it is simpler and computationally faster to use inner products than to use misorientation angles. This is illu-strated neatly in the cubic case. Consider [U] and [V] in SO(3)/O with u 1 , u 2 , u 3 and v 1 , v 2 , v 3 as the columns of U and V, respectively. Then whereas the misorientation angle is the minimum being over all 1 , 2 , 3 = AE 1 and all permutations j, k, ' of 1, 2, 3. Evaluating (41) involves about six times as many operations as evaluating (40).  Table 3 Symmetry groups and frames.
The u i are unit vectors. |K| is the number of elements in K.